1

Manufacturing Planning and Control

Stephen C. GravesMassachusetts Institute of Technology

November 1999

Manufacturing planning and control entails the acquisition and allocation of limitedresources to production activities so as to satisfy customer demand over a specified timehorizon. As such, planning and control problems are inherently optimization problems,where the objective is to develop a plan that meets demand at minimum cost or that fillsthe demand that maximizes profit. The underlying optimization problem will vary due todifferences in the manufacturing and market context. This chapter provides a frameworkfor discrete-parts manufacturing planning and control and provides an overview ofapplicable model formulations.

2

Manufacturing Planning and Control

Stephen C. GravesMassachusetts Institute of Technology

November 1999

Manufacturing planning and control address decisions on the acquisition, utilization andallocation of production resources to satisfy customer requirements in the most efficientand effective way. Typical decisions include work force level, production lot sizes,assignment of overtime and sequencing of production runs. Optimization models arewidely applicable for providing decision support in this context.

In this article we focus on optimization models for production planning for discrete-parts,batch manufacturing environments. We do not cover detailed scheduling or sequencingmodels (e. g., Graves, 1981), nor do we address production planning for continuousprocesses (e. g., Shapiro, 1993). We consider only discrete-time models, and do notinclude continuous-time models such as developed by Hackman and Leachman (1989).

Our intent is to provide an overview of applicable optimization models; we present themost generic formulations and briefly describe how these models are solved. There is anenormous range of problem contexts and model formulations, as well as solutionmethods. We make no effort to be exhaustive in the treatment herein. Rather, we havemade choices of what to include based on personal judgment and preferences.

We have organized the article into four major sections. In the first section we present aframework for the decisions, issues and tradeoffs involved in implementing anoptimization model for discrete-part production planning. The remaining three sectionspresent and discuss three distinct types of models. In the second section we discuss linearprogramming models for production planning, in which we have linear costs. Thiscategory is of great practical interest, as many important problem features can becaptured with these models and powerful solution methods for linear programs arereadily available. In the third section, we present a production-planning model for asingle aggregate product with quadratic costs; this model is of historical significance as itrepresents one of the earliest applications of optimization to manufacturing planning. Inthe final section we introduce the multi-item capacitated lot-size problem, which ismodeled as a mixed integer linear program. This is an important model as it introduceseconomies of scale in production, due to the presence of production setups.

Framework

There are a variety of considerations that go into the development and implementation ofan optimization model for manufacturing planning and control. In this section wehighlight and comment upon a number of key issues and questions that should beaddressed. Excellent general references on production planning are Thomas and McClain(1993), Shapiro (1993) and Silver et al. (1998).

3

Any planning problem starts with a specification of customer demand that is to be met bythe production plan. In most contexts, future demand is at best only partially known, andoften is not known at all. Consequently, one relies on a forecast for the future demand. Tothe extent that any forecast is inevitably inaccurate, one must decide how to account foror react to this demand uncertainty. The optimization models described in this article treatdemand as being known; as such they must be periodically revised and rerun to accountfor forecast updates.

A key choice is what planning decisions to include in the model. By definition,production-planning models include decisions on production and inventory quantities.But in addition, there might be resource acquisition and allocation decision, such asadding to the work force and upgrading the training of the current work force.

In many planning contexts, an important construct is to set a planning hierarchy.Namely, one structures the planning process in a hierarchical way by ordering thedecisions according to their relative importance. Hax and Meal (1975) introduced thenotion of hierarchical production planning and provide a specific framework for this,whereby there is an optimization model with each level of the hierarchy. Eachoptimization model imposes a constraint on the model at the next level of the hierarchy.Bitran and Tirupati (1993) provide a comprehensive survey of hierarchical planningmethods and models.

The identification of the relevant costs is also an important issue. For productionplanning, one typically needs to determine the variable production costs, including setup-related costs, inventory holding costs, and any relevant resource acquisition costs. Theremight also be costs associated with imperfect customer service, such as when demand isbackordered.

A planning problem exists because there are limited production resources that cannot bestored from period to period. Choices must be made as to which resources to include andhow to model their capacity and behavior, and their costs. Also, there may be uncertaintyassociated with the production function, such as uncertain yields or lead times. Onemight only include the most critical or limiting resource in the planning problem, e. g., abottleneck. Alternatively, when there is not a dominant resource, then one must model theresources that could limit production. We describe in this article two types of productionfunctions. The first assumes a linear relationship between the production quantity and theresource consumption. The second assumes that there is a required fixed charge or setupto initiate production and then a linear relationship between the production quantity andresource usage.

Related to these choices is the selection of the time period and planning horizon. Theplanning literature distinguishes between “big bucket” and “small bucket” time periods.A time period is a big bucket if multiple items are typically produced within a timeperiod; a small bucket is such that at most one item would be produced in the time period.For big bucket models, one has to worry about how to schedule or sequence theproduction runs assigned to any time period. The choice of planning horizon is dictated

4

by the lead times to enact production and resource-related decisions, as well as thequality of knowledge about future demand.

Planning is typically done in a rolling horizon fashion. A plan is created for the planninghorizon, but only the decisions in the first few periods are implemented before a revisedplan is issued. Indeed, as noted above, the plan must be periodically revised due to theuncertainties in the demand forecasts and production. For instance a firm might plan forthe next 26 weeks, but then revise this once a month to incorporate new information ondemand and production.

Production planning is usually done at an aggregate level, for both products andresources. Distinct but similar products are combined into aggregate product families thatcan be planned together so as to reduce planning complexity. Similarly productionresources, such as distinct machines or labor pools, are aggregated into an aggregatemachine or labor resource. Care is required when specifying these aggregates to assurethat the resulting aggregate plan can be reasonably disaggregated into feasible productionschedules.

Finally for complex products, one must decide the level and extent of the productstructure to include in the planning process. For instance, in some contexts it is sufficientto just plan the production of end items; the production plan for components andsubassemblies is subservient to the master production schedule for end items. In othercontexts, planning just the end items is sub-optimal, as there are critical resourceconstraints applicable to multiple levels of the product structure. In this instance, a multi-stage planning model allows for the simultaneous planning of end items and componentsor subassemblies. Of course, this produces a much larger model.

5

Production Planning: Linear Programming Models

In this section we develop and state the most basic optimization model for productionplanning for the following context:

• multiple items with independent demand• multiple shared resources• big-bucket time periods• linear costs.

We define the following notation

decision variablespit production of item i during time period tqit inventory of item i at end of time period t

parametersT, I, K number of time periods, items, resources, respectively

aik amount of resource k required per unit of production of item ibkt amount of resource k available in period tdit demand for item i in period t

cpit unit variable cost of production for item i in time period tcqit unit inventory holding cost for item i in time period t

We now formulate the linear program P1:

The objective function (1) minimizes the variable production costs plus the inventoryholding costs for all items over the planning horizon of T periods.

Equation (2) is a set of inventory balance constraints that equate the supply of an item ina period with its demand or usage. In any period, the supply for an item is the inventoryfrom the prior period qi,t-1, plus the production in the period pit. This supply can be usedto meet demand in the period dit, or held in inventory as qit. As we require the inventory

P1: Min (1)

, (2)

, (3)

,

i=1

I

t=1

T

i=1

I

cp p cq q

s t

q p q d i t

a p b k t

p q i t

it it it it

i t it it it

ik it kt

it it

∑∑

∑

+

+ − = ∀

≤ ∀

≥ ∀

−

. .

,

, 1

0

6

to be non-negative, these constraints assure that demand is satisfied for each item in eachperiod. We are given as input the initial inventory for each item, namely qi0.

Equation (3) is a set of resource constraints. Production in each period is limited by theavailability of a set of shared resources, where production of one unit of item i requiresaik units of resource k, for k = 1, 2, ... K. Typical resources are various types of labor,process and material handling equipment, and transportation modes.

The number of decision variables is 2IT, and the number of constraints is IT + KT. Forany realistic problem size, we can solve P1 by any good linear-programming algorithm,such as the simplex method.

We briefly describe next a number of important extensions to this basic model. Weintroduce these as if they were independent; however, we note that many contexts requirea combination of these extensions.

Demand Planning: Lost Sales

For some problems we have the option of not meeting all demand in each time period.Indeed, there might not be sufficient resources to meet all demand. In effect, the demandparameters represent the demand potential, and the optimization problem is to decidewhat demand to meet and how. We assume that demand that cannot be met in a period islost, thus reducing revenue. In addition, a firm might incur a loss of customer goodwillthat would manifest itself in terms of reduced future sales. This lost sales cost is verydifficult to quantify as it represents the future unknown impact from poor service today.

We pose a new planning problem to maximize revenues net of the production, inventoryand lost sales costs. We introduce additional notation and then state the model:

decision variablesuit unmet demand of item i during time period t

parametersrit unit revenue for item i in period tcuit unit cost of not meeting demand for item i in time period t

P2: Max

(3)

,

,

i 1

I

t=1

T

=

−

∑∑ − − − −

+ − + = ∀

≥ ∀

r d u cp p cq q cu u

s t

q p q u d i t

p q u i t

it it it it it it it it it

i t it it it it

it it it

( )

. .

, ,, 1

0

7

The objective function has been modified to include revenue as well as the cost of lostsales. The potential revenue, Σ Σ rit dit, is a constant and could be dropped in theobjective function. In this case, we can restate the problem as a cost minimizationproblem, where the cost of lost sales includes the lost revenue.

Also, in P2, the inventory balance constraint has been modified to permit the option ofnot meeting demand; thus demand in a period can be met from production or inventory,or not satisfied at all. The resource constraint (3) remains unchanged.

Demand Planning: Backorders

A related problem variation is when it is possible to reschedule or backorder demand.That is, we might defer current demand until a later period, when it can be served fromproduction. Of course there is a cost for doing this, which we term the backorder cost.Like the lost sales cost, the backorder cost includes hard-to-quantify costs due to loss ofcustomer goodwill, as well as reduced revenue and additional processing or expeditingcosts due to the deferral of the demand fulfillment. We assume that this cost is linear inthe number of backorders in each period.

We define additional notation and then state the model.

decision variablesvit backorder level for item i at end of time period t

parameterscvit unit cost of backorder for item i in time period t

In comparison with P1, we now include a backorder cost on the objective function for P3.The inventory balance equation is modified to account for the backorders, which in effectbehave like negative inventory. We typically would add a terminal constraint on thebackorders at the end of the planning horizon; for instance, we might require viT = 0, sothat over the T-period planning horizon all demand is eventually met by the productionplan. Any initial backorders, namely vi0, can be dropped by adding them to the first-period demand; that is, we restate the demand as di1 : = di1 + vi0, and then drop vi0 fromthe formulation.

P3: Min

(3)

,

,

i 1

I

t=1

T

cp p cq q cv v

s t

q v p q v d i t

p q v i t

it it it it it it

i t i t it it it it

it it it

+ +

− + − + = ∀

≥ ∀

=

− −

∑∑. .

, ,, ,1 1

0

8

In this formulation, when demand is backordered, the cost of this event is linear in thesize and duration of the backorder. That is, if it takes n time periods to fill the backorder,the cost is proportional to n. In contrast, in some cases, the backorder cost might notdepend on the duration but only on the occurrence and size of the backorder. We canmodify this formulation for this case by defining a variable to represent new backordersin period t, given by max [0, vit - vi,t-1]; then we would apply the backorder cost to thisvariable in the objective function. This modification assumes that we fill backorders in alast-in, last-out fashion, as there is no incentive to do otherwise for this cost assumption.

Piecewise Linear Production Cost Functions

In P1 the relevant production cost is linear in the production quantity. In many contextsthe actual cost function is non-linear. In this section, we consider a convex cost function,and assume that it is well modeled as a piecewise linear function. We model concavecost functions that exhibit economies of scale in alter section.

Let Cost(pit) denote the cost function for item i in period t; we present the case where thiscost function is the same in each period, and introduce the following notation:

decision variablespist production of item i during time period t, that falls in cost segment s

parametersS number of segments in cost functioncpist unit variable cost of production for item i in time period t in cost segment sPis upper bound on cost segment s for item i

Thus, we assume that Cost(pit) is given by:

In order for Cost(pit) to be convex, we require that the unit variable costs be strictlyincreasing from one segment to the next:

0 < cpi1t < cpi2t < ... < cpiSt

This cost function applies when there are multiple options or sources for production, andthese options can be ranked by their variable costs. A common example is when one

Cost p cp p

where

p p

p P s

it iss

S

ist

it ists

S

ist is

( ) =

=

≤ ≤ ∀

=

=

∑

∑

1

1

0

9

models regular time and overtime production. We have two cost segments (S=2) wherethe first segment corresponds to production during regular time, and the second isovertime production. The variable production cost is usually more during overtime, asworkers earn a rate premium. Another example is when the firm works multiple shiftsand the variable costs differ between these shifts. A third example is when there aresubcontracting or outsourcing options; there are multiple costs segments, one to representin-house production and the others to represent the outsourcing options ranked by cost.

We model the planning problem with convex, piecewise linear production cost functionsby replacing the production cost in P1 with the above formulation for Cost(pit):

In P4 we have modified the resource constraints (3) to accommodate the possibility thatthe usage of the shared resources depends on the production quantity by source or costsegment, i. e., pist, rather than just on pit. In this case, aiks denotes the amount of resourcek required per unit of item i produced at source or cost segment s. This form permits greatflexibility in modeling production costs as well as resource constraints.

In one of the first papers on production planning, Bowman (1956) formulates thisproblem as a transportation problem, when there are multiple time periods and multipleproduction options, but only one item and one resource type.

Resource Planning

Up to now we have assumed that the resource levels are fixed and given. In some cases,an important element of the planning problem is to decide how to adjust the resourcelevels over the planning horizon. For instance, one might be able to change the workforce level, by means of hiring and firing decisions. Hansmann and Hess (1960) providean early example of this type of model.

Suppose for ease of notation that we have just one type of resource, namely the workforce. We introduce additional notation and then state the model:

decision variableswt work force level in time period t

P4: Min )

(2)

,

, ,

i=1

I

t=1

T

i=1

I

(

. .

,

,

, ,

cq q cp p

s t

a p b k t

p p i t

p P i s

p p q i s t

it it iss

S

ist

iks ist kt

it ists

S

ist is

it ist it

∑∑ ∑

∑

∑

+

≤ ∀

− = ∀

≤ ≤ ∀≥ ∀

=

=

1

1

0

0

0

10

ht change to work force level by hiring in time period tft change to work force level by firing in time period t

parametersai amount of work force (labor) required per unit of production of item icwt variable unit cost of work force in time period tcht variable hiring cost in time period tcft variable firing cost in time period t

We add the variable cost for the work force to the objective function, along with costs forhiring and firing workers. The hiring cost includes costs for finding and attractingapplicants as well as training costs. The firing cost includes costs of outplacement andretraining of displaced workers, as well as severance costs; there might also be a cost oflower productivity due to lower work-force morale, when firings or layoffs occur.

The inventory balance constraint (2) remains the same as for P1, and we restate theresource constraint, reflecting the work force as a decision variable and as the soleresource. We then add a new set of balance constraints for planning the work force: thework force in period t is that from the prior period plus new hires minus the number fired.

We have stated P5 for a single resource, representing the work force. The model extendsimmediately to include other resources that might be managed in a similar fashion overthe planning horizon. In addition, there might be other considerations to model such astime lags when adjusting a resource level. There might be limits on how quickly newworkers can be added due to training requirements. If there were limited trainingresources, then this imposes a constraint on ht. Alternatively, new hires might be lessproductive until they have acquired some experience. In this case, we modify theformulation to model different categories of workers, depending on their tenure andexperience level.

Another common variation of this model is when there are two labor classes, say,permanent employees and temporary employees. These classes differ in terms of theircost coefficients, and possibly their efficiency factors. Permanent employees have higherhiring and firing cost, as they receive more training and have more rights and protectionfrom layoffs. But their variable production cost, normalized by their productivity, shouldbe lower than that for temporary workers. The planning problem then entails the

P5: Min

(2)

0

,

t=1

T

i 1

I

t 1

T

i 1

I

cw w ch h cf f cp p cq q

s t

a p w t

w h f w t

p q w h f i t

t t t t t t it it it it

i it t

t t t t

it it t t t

+ + + +

− ≤ ∀

+ − − = ∀≥ ∀

∑ ∑∑

∑

==

=

−

. .

, , , ,1 0

0

11

management and planning of both work classes over the planning horizon. Forcompleteness, we revise P5 for two work force classes:

decision variableswjt work force level in time period thjt change to work force level by hiring in time period tfjt change to work force level by firing in time period tpijt production of item i during time period t, using labor class j

parametersaij amount of labor required per unit of production of item i, using labor class jcwjt variable unit cost of labor class j in time period tchjt variable hiring cost for labor class j in time period tcfjt variable firing cost for labor class j in time period t

In comparison to P5, we have decision variables for both labor classes, as well as for theirhiring and firing decisions, in order to model the cost differences. We also introduceproduction decision variables, by labor class, to capture the differences in productivity.

Multiple Locations

In P1, there is a single supply location or production facility that serves demand for allitems. Often there are multiple production facilities that are geographically dispersed andthat supply multiple distribution channels. The planning problem is to determine theproduction, inventory and distribution plans for each facility to meet demand, which isnow modeled by geographic region.

There are many ways to formulate this type of problem. We provide an example, andthen comment on some variants. We define the notation and then state the model.

P6: Min

(2)

,

0

, ,

j=1

2

t=1

T

i 1

I

t 1

T

j 1

2

i 1

I

cw w ch h cf f cp p cq q

s t

p p i t

a p w j t

w h f w j t

p q w h f i j t

jt jt jt jt jt jt it it it it

it ijt

ij ijt jt

j t jt jt jt

it it jt jt jt

+ + + +

− = ∀

− ≤ ∀

+ − − = ∀

≥ ∀

∑∑ ∑∑

∑

∑

==

=

=

−

. .

,

,

, , , ,,

0

0

01

12

decision variablespijt production of item i at facility j during time period tqijt inventory of item i at facility j at end of time period txijmt shipment of item i from facility j to demand location m in time period t

parametersT, I, K number of time periods, items, resources, respectivelyJ, M number of facility locations, demand locations, respectively

aijk amount of resource k required per unit of production of item i at facility jbjkt amount of resource k available at facility j in period tdimt demand for item i at location m in period t

cpijt unit variable cost of production for item i at facility j in time period tcqijt unit inventory holding cost for item i at facility j in time period tcxijmt unit transportation cost to ship item i from facility j to demand location m in time

period t

The decision variables for production and inventory are now specified by location, whereinventory is held at the production locations. In addition, we have aggregated demandinto a set of demand regions or locations, and introduce a new set of decision variables todenote transportation from the production facilities to the demand locations. Theobjective function captures production and inventory holding cost, which depends on thefacility, plus transportation or distribution costs for moving the product from a facility tothe demand location. The inventory balance constraints assure that the supply of an itemat each facility is either held in inventory or shipped to a demand location to meetdemand. The second set of constraints assures that the shipments satisfy the demandeach period at each location. The resource constraints are structurally the same as in P1,but for multiple production locations.

A key variant of this model occurs when there are additional stocking locations, such as anetwork of warehouses or distribution centers. These stocking locations not only provide

P7: Min

,

,

, ,

, , ,

j=1

J

i=1

I

m 1

M

t=1

T

m 1

M

j 1

J

i=1

I

[ ]

. .

,

,

, ,

,

cp p cq q cx x

s t

q p q x i j t

x d i m t

a p b j k t

p q x i j m t

ijt ijt ijt ijt ijmt ijmt

ij t ijt ijt ijmt

ijmt imt

ijk ijt jkt

ijt ijt ijmt

∑∑ ∑∑

∑

∑

∑

+ +

+ − − = ∀

= ∀

≤ ∀

≥ ∀

=

−=

=

1 0

0

13

additional space to store inventory close to the demand locations, but also permiteconomies of scale in transportation from the production sites to these stocking locations.In this case, one defines shipments to and from each stocking location, and has inventorybalance constraints for each stocking location. The shipment costs would capture anydifferences in transportation modes that might be employed.

The size of P7 creates a challenge for implementing and maintaining such a model.There are now IJT(2+M) decision variables and (IJT + IMT + JKT) constraints. Atypical problem might have 20 – 100 aggregate item families, 5 - 10 facilities, 50 – 100demand locations, 12 – 20 time periods, and 1 – 5 resource types. Thus, the model mighthave on the order of 100,000 to 1 million decision variables, and on the order of 100,000constraints. As the problem is still a linear program, such problems are readily solved bycommercial optimization packages. However, the real difficulty in such implementationsis the development and maintenance of the parameters. There can be on the order of onemillion demand forecasts, 100,000 to 1 million cost coefficients, and 10,000 resourcecoefficients. The success of many applications often rests on whether this data can beobtained and kept accurate.

Dependent Demand Items

So far we have considered production plans for end items or finished goods, which seeindependent demand. But these end items are usually comprised of many fabricatedcomponents and subassemblies, for which there needs to be a production plan too. Theseitems differ from the end items, in that their demand depends on the end-item productionplan. Material requirements planning (MRP) systems are designed to characterize thisdependent demand and to facilitate the planning for these dependent-demand items in acoordinated and systematic way (Vollman et al. 1992). Nevertheless, it is quite easy toincorporate dependent-demand items into the optimization models presented here.

W first need to define the “goes into” matrix A = {αij}, where αij is the number of unitsof item i required per unit of production of item j. Then for the model P1 we replace theinventory balance constraints (2) with the following:

In this general form, the demand for item i has two parts: exogenous demand given by dit,and endogenous or induced demand given by Σαij pjt. For end items we expect there to beno induced demand, whereas for components, there will typically be no or limitedexogenous demand.

One variation to this model is when there are manufacturing lead times, wherebyproduction of item j in time period t requires that component i be available at time t – L,where L is the lead time for producing j. The inventory balance constraints can be easilymodified to accommodate this, given that the lead times are known and deterministic.

q p q p d i ti t it it ij jt it, −=

+ − − = ∀∑1 αj 1

I

,

14

Billington et al. (1983) provide a comprehensive treatment of this model and problem,and present methods for reducing the size of the problem so as to facilitate its solution. Inthe literature, this problem is referred to as a multiple stage problem, where end items,subassemblies, and components might represent distinct stages in a manufacturingprocess.

15

Quadratic Cost Models and Linear Decision Rule

One of the earliest production-planning modeling efforts was that of Holt, Modigliani,Muth and Simon (1960), who developed a production-planning model for the PittsburghPaint Company. They assume a single aggregate product, and then define three decisionvariables:

pt production of the aggregate item during time period tqt inventory of the aggregate item at end of time period twt work force level in time period t

Holt, Modigliani, Muth and Simon (HMMS) assume that the cost function in each periodhas four components. The first component is the regular payroll costs that is a linearfunction of the work force level. The second component is the hiring and layoff costs,which were assumed to be a quadratic function in the change in work force from oneperiod to the next.

The next cost component is for overtime and idle-time costs. HMMS assume there is anideal production target that is a linear function of the work-force level. If production isgreater than this target there is an overtime cost, while if production is less than thistarget there is an idle-time cost. Again, HMMS assume that the cost is quadratic aboutthe variance between the actual production and the production target for the work-forcelevel.

The final component is inventory and backorder costs. Similar to the overtime and idle-time costs, there is an inventory target each period, which is a linear function of thedemand in the period. The inventory and backorder cost is a quadratic function of thedeviation between the inventory and the inventory target.

The HMMS optimization is to minimize the sum of the expected costs over a fixedhorizon, subject to an inventory balance constraint. The expectation is over the demandrandom variables, where we are given an unbiased forecast of demand over the planninghorizon. The analysis of this optimization yields two key results.

First, the optimal solution can be characterized as a linear decision rule, whereby theaggregate production rate in each period is a linear function of the future demandforecasts, as well as the work force and inventory level in the prior period. There is asimilar linear function for specifying the work force level in each period.

Second, the optimal decision rule is derived for the case of stochastic demand, but onlydepends on the mean of the demand random variables. That is, we only need to know (orassume) that the demand forecasts are unbiased in order to apply the linear decision rule.

16

Production Planning: Lot-Size Models

In this section we consider production-planning problems for which there are economiesof scale associated with the production activity or function. The most common exampleoccurs when there is a required setup to initiate the production of an item. For instance,to initiate the production of an item might require a change in tooling, or dies, or rawmaterial; the setup might also require a change in the production control settings, as wellas an initial run to assure that the production output meets quality specifications. Theremight be a setup cost, corresponding to labor costs for performing the setup, plus directexpenditures for materials and tools. There might also be resource requirements for thesetup, usually referred to as the setup time. The production resource cannot produce untilthe setup is completed; thus the setup consumes production capacity, equal to itsduration, namely the setup time.

Given the presence of setups, once an item is setup to produce, we may want to produce alarge batch or lot size so as to cover demand over a number of future periods and hencedefer the next time when the item will be setup and produced. Whereas producing largebatches will reduce the setup costs, this also increases inventory as more demand isproduced earlier in time. The lot-sizing problem, as described here, is to determine therelative frequency of setups so as to minimize the setup and inventory costs, within theresource and service constraints of the production-planning problem.

We start with the simplest model and then briefly discuss variants to it. We develop andstate this model for the following context:

• multiple items with independent demand• a single shared resource• big-bucket time periods• linear costs, except for setup costs.

For ease of presentation we assume a single resource; the extension to consider multipleresources is straightforward.

We define the following notation

decision variablespit production of item i during time period tqit inventory of item i at end of time period tyit binary decision variable to denote setup of item i in time period t

parametersT, I number of time periods, items, respectively

ai1 amount of resource required per unit of production of item iai2 amount of resource required for setup of production of item i

17

bt amount of resource available in period tdit demand for item i in period tB a large constant

cpit unit variable cost of production for item i in time period tcqit unit inventory holding cost for item i in time period tcyit setup cost for production for item i in time period t

We now formulate the mixed-integer linear program P8:

The objective function (4) minimizes the sum of variable production costs, the inventoryholding costs and the setup costs for all items over the planning horizon of T periods.

Equation (5) is the same as (2), the inventory balance constraints that equate the supply ofan item in a period with its demand or usage.

The resource constraints (6) reflect the resource consumption both due to the productionquantity for each item, and due to the setup. Production of one unit of item i consumes ai1

units of the shared resource, while the setup requires ai2 units.

The constraint set (7) is for the so-called forcing constraints. These constraints relate theproduction variables to the setup variables. For each item and time period, if there is nosetup (yit=0), then this constraint assures that there can be no production (pit=0).Conversely, if there is production in a period (pit>0), then there must also be a setup(yit=1). In (7), B is any large positive constant that exceeds the maximum possible valuefor pit; for instance, one might set B equal to the sum of all demand.

This problem is now a mixed-integer linear program, with IT binary decision variables.For modest size problems with, say, a few hundred binary decision variable, this problemcan be reliably solved by commercial optimization packages. But specialized approachesare warranted for increasing problem size and complexity. We discuss one of theseapproaches next.

P8: Min (4)

, (5)

(6)

, (7)

= , ,

i=1

I

t=1

T

i=1

I

cp p cq q cy y

s t

q p q d i t

a p a y b t

p By i t

p q y i t

it it it it it it

i t it it it

i it i it t

it it

it it it

∑∑

∑

+ +

+ − = ∀

+ ≤ ∀

≤ ∀≥ ∀

−

. .

, ;

, 1

1 2

0 0 1

18

Dual-Based Solutions

In this section we develop a dual problem for P8. We identify a generalized linearprogram for solving this dual problem, which is equivalent to a convexification of P8. Wecan solve this problem by column generation to obtain a lower bound on P8; we alsodiscuss how this solution can be used to identify near optimal solutions to P8.

The approach is based on the original work of Manne (1958) who suggested thegeneralized linear program given below. Dzielinski and Gomory (1965) extended themodel of Manne to include resource planning decisions (i. e., hiring and firing labor) andapplied the Dantzig-Wolfe decomposition method to introduce column generation.Lasdon and Terjung (1971) reformulated the linear program so as to provide a moreefficient and effective column generation approach for solving the linear program; theyalso address a variant of P8, where there are small time buckets for planning productionand multiple resources corresponding to scarce machines and dies.

To develop the dual problem, we first define a Lagrangean function L(π) by dualizing theresource constraints (6):

where π = (π1, π2, ... πT) ≥ 0 is a vector of dual variables.

We state the following observations:

• The Lagrangean separates by item, where we have a single-item dynamic lot-sizeproblem with no capacity constraints for each item. This is the so-called Wagner-Whitin (1958) problem and can be solved by dynamic programming.

• If qi0 = 0, the extreme points to L(π), namely the single-item dynamic lot-sizeproblem, have the property whereby pit qit-1=0. As a consequence the optimalproduction quantities are a sum of consecutive demands. That is, if pit > 0, then pit =dit + ... + dis for t ≤ s ≤ T. We will refer to this as a “Wagner-Whitin schedule.”

• Without loss of generality, we will assume that qi0 = 0, and the above propertyapplies to optimal solutions. If qi0 > 0, then we use this initial inventory to reduce theitem’s demand. That is, we restate the demand as follows:

dit : = 0 for t = 1, 2, … s-1dis : = di1 + di2 + … + di, s-1 - qi0

where s such that di1 + di2 + … + di, s-1 ≤ qi0 < di1 + di2 + … + dis .

We can use the Lagrangean function to define a dual problem to P8:

L b cp a p cq q cy a y

s t

p q y i t

t t it i t it it it it i t it

it it it

( ) = Min -

= , ,

t=1

T

i=1

I

t=1

T

π π π π∑ ∑∑+ + + + +

≥ ∀

( ) ( )

. .( ),( )

, ;

1 2

5 7

0 0 1

19

The dual solution need not and usually will not identify a primal feasible solution to P8.In such instances, a duality gap exists and the dual solution provides a lower bound to P8.One might consider two procedures for resolving the duality gap. First, one could use thedual problem for generating bounds in a branch and bound procedure; the effectivenessof this depends on the tightness of the bounds and the number of integer variables in P8.The second approach incorporates the solution of the dual problem into a heuristicprocedure. For each iteration in the solution of the dual, we might generate, somehow, acorresponding feasible solution to P8. The best such feasible solution can be comparedwith the solution of the dual to assess its near optimality; the procedure stops once thebest feasible solution is sufficiently close to the optimum or after a predeterminednumber of iterations.

We might solve the dual D8 directly by means of a method such as subgradientoptimization, or a dual-ascent procedure. Alternatively, we can follow the generalderivation provided by Magnanti et al. (1976) to reformulate the dual problem as ageneralized linear program. We denote the extreme points of the convex hull defined byconstraint sets (5) and (7), the non-negativity constraints and the binary constraints by

where for each item i, uij is a Wagner-Whitin schedule.

We define the cost for the jth extreme point and resource usage for the jth extreme point intime period t as

We can rewrite D8 in terms of the extreme points as the following equivalent linearprogram:

where z is unconstrained in sign. The dual of this problem is:

D Max L

s t

8: ( )

. .

0

ππ ≥

u u u u p q y p q yj j jIj

tj

tj

tj

I tj

I tj

I tj= =( , ,... ) (( , , ),...( , , )),, , ,1 2 1 1 1

cu cp p cq q cy y

a a p a y

jit it

jit it

jit it

j

tj

i itj

i itj

= + +

= +

∑∑

∑= t=1

T

i 1

I

i=1

I

1 2 .

Max z

s t

z b a cu j

t

t tj

tt

Tj

t

. .

( )

,

+ − ≤ ∀

≥ ∀=

∑ π

π1

0

20

where J is the number of extreme points; the decision variable xj indicates the fraction ofthe schedule given by the jth extreme point. Problem P9 is a convexification of the primalproblem P8 in which we replace the feasible region defined by constraints (5), (7), thenon-negativity and binary constraints by the convex hull of this region. The solution ofP9 provides a lower bound for P8. However, P9 is not all that useful due to the largenumber of variables, on the order of 2IT; and the solution to P9 will typically befractional, and not suggestive of good, near-optimal feasible solutions.

We can reformulate P9 by noting that we can express the set of extreme points U = {uj}in terms of extreme points for the individual items. That is,

U = U1 × U2 × ... × UI

where

is the set of extreme points or Wagner-Whitin schedules for item i. For the kth extremepoint for item i, we define its cost and resource usage parameters:

We now can rewrite P9 in terms of the Wagner-Whitin schedules for the items:

P Min cu x

s t

x

a x b t

x j

jj

j

tj

j t

j

9

1

0

:

. .

,

j 1

J

j=1

J

j 1

J

=

=

∑

∑

∑

=

≤ ∀

≥ ∀

Ui iku= { }

cu cp p cq q cy y

a a p a y

ik

it itk

it itk

it itk

itk

i itk

i itk

= + +

= +

∑t=1

T

1 2 .

21

where Ki denotes the number of Wagner-Whitin schedules for item i and the decisionvariable xik is the fraction of the kth Wagner-Whitin schedule for item i in the solution.The first constraint assures that these fractions sum to one, so that the demand for eachitem is met. The second constraint enforces the resource constraint.

The optimal solution to P10 solves the dual D8, and provides a lower bound to P8. If theoptimal solution to P10 is all integer (xik = 0 or 1 for all i, k ), then this solution is optimalto P8. When the solution to P10 is not integer, it provides the basis for finding nearoptimal solutions. We describe this solution strategy in two parts: first how we solve P10and second how we use the solutions to P10 to obtain good feasible solutions to P8.

Manne (1958) first proposed solving P10 as an approximation to solving P8. However, aswith P9, there can be an enormous number of decision variables, on the order of I2T inthis case. Rather than generate all of these decision variables and their parameters, wesolve P10 by means of column generation [Dzielinski and Gomory (1965), Lasdon andTerjung (1971)]. At each iteration, we solve a master problem, namely a reduced versionof P10 with a subset of columns. Then, using the dual values from the master problem,we solve a Wagner-Whitin problem for each item to find a new candidate schedule toenter into the master problem. The procedure iterates between the master problem and theitem sub-problems until the solutions to the Wagner-Whitin sub-problems yield no newitem schedules. One would typically terminate this procedure when the gap between thelower bound provided by the master problem and a known feasible solution to P8 issuitably small.

Manne (1958) noted that although solutions to P10 are usually fractional, there is aninteger solution for most items. An optimal solution to P10, as well as the master problemin the column generation procedure, has I+T basic variables, as there are I+T constraints.Thus, there are at most I+T fractional variables in the solution. Each item must have atleast one basic variable, in order to satisfy the convexity constraint in P10. As aconsequence, no more than T items can have two or more basic variables. Thus, at least I-T items have a single basic variable, which must be integer due to the convexityconstraint.

For most planning problems the number of items would be much greater than the numberof time periods. For instance, we might be planning for 50 – 100 items, with 12 or 13

P Min cu x

s t

x i

a x b t

x i k

ik

ik

ik

itk

ik t

ik

10

1

0

:

. .

, ,

k 1

K

i=1

I

k 1

K

k 1

K

i 1

I

i

i

i

=

=

==

∑∑

∑

∑∑

= ∀

≤ ∀

≥ ∀

22

time periods. For I=100 and T=13, a solution to P10 provides a single Wagner-Whitinschedule for at least 87 items. For the remaining items, the solution suggests a convexcombination of two or more Wagner-Whitin schedules. These schedules satisfy thedemand constraints in P8, but require more setup time and cost than assumed by P10.[P10 assumes that we only incur a fraction of the setup cost and time, for a schedule at afractional level of activity.] Thus, the set of schedules from P10 might violate theresource constraints in P8. Manne notes that in some contexts, the resource constraintsare soft constraints and minor violations can be ignored. In other cases, one might applya heuristic to modify the solution from P10 so as to make it satisfy the resourceconstraints. One expects that it is relatively easy to find a feasible solution to P8 from thesolution to P10, as most of the items have a single schedule. Furthermore, one expectsthat the feasible solutions are near optimal, again due to the fact that the solution to P10 isnear integer. Computational experience in Manne, Dzielinski and Gomory and Lasdonand Terjung supports these claims. Also, Trigeiro et al. (1989) describe and test aheuristic smoothing procedure for constructing feasible solutions.

Variable Redefinition

Eppen and Martin (1987) report on an alternative solution approach to P8, based onvariable redefinition. They reformulate P8 so that its LP relaxation provides a very tightbound, comparable to that from the dual-based approaches of Lagrangean relaxation orcolumn generation. They then solve the mixed integer program using general-purposeoptimization codes to obtain optimal or near-optimal solutions for problems with up to200 products and ten time periods.

Their approach is based on the optimal property of Wagner-Whitin schedules: when aproduction activity occurs, the production quantity covers the demand in an integernumber of consecutive periods beginning with the period of production. They thendefine decision variables for each such production opportunity for each item. With thesenew variables, the schedule for each item is a shortest path problem through a network ofT nodes. These shortest path problems are coupled by resource constraints that cut acrossthe individual items.

For completeness we present the Eppen-Martin model for multi-item capacitated lot-sizing.

decision variableszitk binary decision variable to denote production of item i during time period t, where

the production quantity is to satisfy demand for periods t through kyit binary decision variable to denote setup of item i in time period t

parametersT, I number of time periods, items, respectively

aitk amount of resource required in period t, if zitk = 1

23

bt amount of resource available in period t

czitk variable production and inventory holding cost for item i, for production in periodt to satisfy demand from period t to k for item i.

cyit setup cost for production for item i in time period t

We now formulate the mixed-integer linear program P11:

The first set of constraints corresponds to the resource constraint (6) in P8. The next twosets model the flow balance for the underlying shortest path problem for each item onnodes 1, 2, ...T, where zitk corresponds to the flow on the arc from node t to node k+1.The last set is the forcing constraint, equivalent to (7) in P8. Note that the demandparameters dit do not appear in P11. Rather they are embedded in the definition of thecost parameters czitk, and the resource coefficients aitk. Also, whereas we define zitk to bea binary variable, we do not make this an explicit requirement when solving P11. Eppenand Martin find that P11 provides a tight lower bound to P8 and also identifies near-optimal feasible solutions.

Extensions to the Lot Size Model

We can extend formulation P8 to incorporate all of the problem variations that wereintroduced for the linear-programming production-planning model. In addition wemention here three other common extensions.

In P8 we assume that any quantity could be produced, subject to the resource constraint.Furthermore, the resource usage for a production quantity consists of a fixed setup timeand a variable amount linear in the production quantity. In some contexts, productionoccurs as a batch process, e. g., a heat treat or diffusion process. Each batch produces afixed amount of product and consumes a fixed amount of the limited resource. We can

P Min cz z cy y

s t

a z b t

z i

z z i t

z y i t

z y i t k

itk itk it it

itk itk t

i k

itk ik t

itk it

itk it

11

1

0

0 0 1

1

1

:

. .

,

,

, , , ,

,

k=t

T

t=1

T

i=1

I

t=1

T

i=1

I

k t

T

i 1

I

k=1

T

k=t

T

k=1

t-1

k=t

T

∑∑∑ ∑∑

∑∑

∑

∑ ∑

∑

+

≤ ∀

= ∀

− = ∀

≤ ∀

≥ = ∀

==

−

24

model this by introducing an integer decision variable for the number of batchesproduced of an item in a time period.

A second variation is when the setups are sequence dependent; that is, the setup for anitem will depend upon what was just previously processed. There is not an easy way tomodify P8 to accommodate this feature. Indeed, in general, the standard representationof sequence-dependent setups is to map this into a traveling salesman problem, whichresults in a new level of complexity.

The third variation is when setups can be carried over from one period to the next. In P8we assume that this is not possible; that is, every period that we produce an item we incura setup. In some contexts, though, we might be able to preserve the last setup in theperiod. Thus, we would incur only one setup if we produce item i last in one period andfirst in the next period. Karmarkar et al. (1987) examine a single-product version of thisproblem. Another variation of this problem is when there are multiple products and weassume small time buckets, so that at most one item is produced in a period. Lasdon andTerjung formulate and address this problem by means of column generation. and Eppenand Martin show how to solve both of these problems efficiently by variableredefinition.

Word Count: 7686

25

References

Billington, P. J., J. O. McClain and L. J. Thomas, “Mathematical Approaches toCapacity-Constrained MRP Systems: Review, Formulation and Problem Reduction,”Management Science, Vol. 29, No. 10 (October 1983), pp.1126-1141.

Bitran, G. R. and D. Tirupati, “Hierarchical Production Planning,” In Handbooks inOperations Research and Management Science, Volume 4, Logistics of Production andInventory, edited by S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin, Amsterdam,Elsevier Science Publishers B. V., 1993, pp. 523-568.

Bowman, Edward H., “ Production Scheduling by the Transportation Method of LinearProgramming,” Operations Research, Vol. 4, No. 1, (February 1956), pp. 100-103.

Dzielinski, B. P. and R. E. Gomory, “Optimal Programming of Lot Sizes, Inventory andLabor Allocations,” Management Science, Vol. 11, No. 9 (July 1965), pp. 874-890.

Eppen, G. D. and R. K. Martin, “Solving Multi-Item Capacitated Lot-Sizing ProblemsUsing Variable Redefinition,” Operations Research, Vol. 35, No. 6 (November-December 1987), pp. 832-848.

Graves, S. C., “A Review of Production Scheduling,” Operations Research, Vol.29, No. 4 (July-August 1981) pp. 646-675.

Hackman, S. T. and R. C. Leachman, “A General Framework for ModelingProduction,” Management Science, Vol. 35, No. 4 (April 1989), pp. 478-495.

Hansmann, F. and S. W. Hess, “A Linear Programming Approach to Production andEmployment Scheduling,” Management Technology, Vol. 1, No. 1, (1960), pp. 46-51.

Hax, A. C. and H. C. Meal, “”Hierarchical Integration of Production Planning andScheduling,” In Studies in Management Sciences, Vol. 1: Logistics, edited by M. A.Geisler, New York, Elsevier, 1975, pp. 53-69.

Holt, C. C., F. Modigliani, J. F. Muth and H. A. Simon, Planning Production, Inventoriesand Work Force, Englewood Cliffs NJ, Prentice-Hall, 1960.

Karmarkar, U. S., S. Kekre and S. Kekre, “The Dynamic Lotsizing Problem with Startupand Reservation Costs,” Operations Research, Vol. 35, No. 3 (May-June 1987), pp. 389-398.

Lasdon, L. S. and R. C. Terjung, “An Efficient Algorithm for Multi-Item Scheduling,”Operations Research, Vol. 19, No. 4 (July-August 1971), pp. 946-969.

Magnanti, T. L, J. F. Shapiro, and M. H. Wagner, “Generalized Linear ProgrammingSolves the Dual,” Management Science, Vol. 22, No. 11 (July 1976), pp. 1195-1203.

26

Manne, A. S., “Programming of Economic Lot Sizes,” Management Science, Vol. 4, No.2 (January 1958), pp. 115-135.

Shapiro, J. F., “Mathematical Programming Models and Methods for ProductionPlanning and Scheduling,” In Handbooks in Operations Research and ManagementScience, Volume 4, Logistics of Production and Inventory, edited by S. C. Graves, A. H.G. Rinnooy Kan and P. H. Zipkin, Amsterdam, Elsevier Science Publishers B. V., 1993,pp. 371-443.

Silver, E. A., D. F. Pyke, and R. Peterson, Inventory Management and ProductionPlanning and Scheduling, 3rd Edition, New York, John Wiley Inc., 1998.

Thomas. L. J. and J. O. McClain, “An Overview of Production Planning,” In Handbooksin Operations Research and Management Science, Volume 4, Logistics of Production andInventory, edited by S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin, Amsterdam,Elsevier Science Publishers B. V., 1993, pp. 333-370.

Trigeiro, W. W., L. J. Thomas and J. O. McClain, “Capacitated Lot Sizing with SetupTimes,” Management Science, Vol. 35, No. 3 (March 1989), pp. 353–366.

Vollman, T. E., W. L. Berry and D. C. Whybark, Manufacturing Planning and ControlSystems, 3rd edition, Burr Ridge Ill., Richard D. Irwin Inc., 1992.

Wagner, H. M. and T. Whitin, “Dynamic Version of the Economic Lot Size Model,”Management Science, Vol. 5, No. 1 (October 1958), pp. 89-96.